Find the sum of $8 + 11 + 14 +... + 602 + 605$.
Getting started We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $3$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {8})$ and the last term $(a_n = {605})$ are given in the question. We need to find $n$ (the number of terms). Step 1: Find $n$ (the number of terms) The sequence increases by $605 - 8 = 597$ from the first term to the last term. Because the sequence increases by $3$ each time, it takes $\dfrac{597}{3} = 199$ terms to get from the first term to the last term. We still need to count the first term, so there are $199 + 1 = {200}$ terms in the sequence. In other words, $n = {200}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{200}}&= \dfrac {\left({8} + {605} \right)}{2} \cdot {200} \\\\ S_{{200}} &= 306.5 \left(200\right) \\\\ S_{{200}} &= 61{,}300\end{aligned}$ The answer $ 61{,}300 $